Natural Density and Logarithmic Density Natural density of a set $S = (a_1,a_2,...) $  is defined (assuming it exists) as
$$\lim_{n \to \infty}\frac{1}{n}\sum\limits_{k\in S, k\le n}1 $$
The logarithmic density of the same set is defined as
$$\lim_{n \to \infty}\frac{1}{\log n}\sum\limits_{k\in S, k\le n}\frac{1}{k} $$
My first question is are there any examples of a set where the natural density does not exist but the logarithmic density does. My second question is, can anyone give rough properties (necessary or sufficient) a set must have in order for its natural density to exist. 
P.S. I am aware that if you have a set with sufficiently large 'blocks' of elements and then sufficiently large 'gaps', its natural density cannot exist. However the task I am trying to complete is to prove that the natural density of a certain set does exist  (without necessarily finding it) so any conditions this set must have is what I am really looking for.
 A: For your first question: consider the set $S=\{2,3, 8,9, ... , 15, 32, ..., 63, ..., 2^{2k-1}, ..., 2^{2k}-1, 2^{2k+1}, ... \}$.
For $2^{2k} \leq N < 2^{2s+1}$ we have
$1/\log N \sum_{n\leq N, n \in S} 1/n=1/\log N \cdot \sum_{j=0}^{k-1} \sum_{\tau=0}^{2^{2j+1} } 1/(2^{2j+1}+\tau)=1/2 + O(1/(\log N))$ and similarly for $2^{2s+1} \leq N < 2^{2s+2}$. So logarithmic density exists.
Natural density does not exist: the density on the interval $1,.., 2^{2k}$ converges to $2/3$ and on the interval $1, ..., 2^{2k+1}$  to $1/3$.
I have never seen the conditions for the existance of natural density. However, there are many theorems that answers the question whether the following limit exists  $\lim_{n \to \infty} 1/n (f(1) + ... +f(n))$ (a natural mean) for some number-theoretic function $f$. Such questions are quite subtle. For example one can approach them using the analytic properties of the Mobius inverse of $f$. The following is true:
If $f(n)=\sum_{d|n}g(d)$ (such $g$ always exists by Mobius inversion) then if the series $\sum_{d=1}^{\infty}g(d)/d$ converges then $f$ has a natural mean. 
You can try to apply these to the characteristic function of the set...
